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Section Polar Coordinates

Like the Cartesian (rectangular) coordinate system that you are familiar with, the polar coordinate system is a way to specify the location of points in the plane. Some curves have simpler equations in polar coordinates, while others have simpler equations in the Cartesian coordinate system.

Example 7.35.

Convert a few points from polar coordinates to rectangular coordinates and vice-versa — perhaps \((3,4)\text{,}\) \((0,1)\text{,}\) \((5\pi/6, 2)\text{,}\) and \((3\pi/2,-3)\text{.}\) Convert a few equations from polar to rectangular coordinates and vice-versa — perhaps \(5x-3y=9\text{,}\) \(3x^2=y\text{,}\) \(r = 2\sin(\theta)\text{,}\) and \(r=4\text{.}\)

Definition 7.36.

Given a point \(P\) in the plane we may associate \(P\) with an ordered pair \((r, \theta)\) where \(r\) is the distance from \((0,0)\) to \(P\) and \(\theta\) is the angle between the \(x-\)axis and the ray \(\overrightarrow{OP}\) as measured in the counter-clockwise direction. The pair \((r, \theta)\) is called a polar coordinate representation for the point \(P\text{.}\)

Warning. There are many polar coordinate representations for a point. For example, if \((r, \theta)\) is a polar coordinate representation for some point in the plane, then by adding \(2\pi\) to the angle, \((r, \theta + 2\pi)\) is a second polar coordinate representation for the same point. We also allow \(r \leq 0\) so that by negating \(r\) and adding \(\pi\) to the angle, \((-r, \theta + \pi)\) is a third polar coordinate representation for the same point.

Problem 7.37.

Plot the following points in the polar coordinate system and find their coordinates in the Cartesian coordinate system.

  1. \((1, \pi)\)

  2. \(\dsp \Big(3, \frac{5 \pi}{4}\Big)\)

  3. \(\dsp \Big(-3, \frac{\pi}{4}\Big)\)

  4. \(\dsp \Big(-2, \frac{- \pi}{6}\Big)\)

Problem 7.38.

Suppose \((r, \theta)\) is a point in the plane in polar coordinates and \((x,y)\) are the coordinates in the Cartesian coordinate system. Write formulas for \(x\) and \(y\) in terms of \(r\) and \(\theta\text{.}\)

Problem 7.39.

Suppose \((x,y)\) is a point in the plane in Cartesian coordinates. Write formulas (in terms of \(x\) and \(y\)) for

  1. the distance \(r\) from the origin to \((x,y)\) and

  2. the angle between the positive x-axis and the line containing the origin and \((x,y)\text{.}\)

Problem 7.40.

Graphing.

  1. Graph the curve \(y = \sin(x)\) for \(x\) in \([0, 2 \pi]\) in the Cartesian coordinate system.

  2. Graph the curve \(r = \sin(\theta)\) for \(\theta\) in \([0, 2\pi]\) in the polar coordinate system.

Problem 7.41.

Each of the following equations is written in the Cartesian coordinate system. Use the relationships you found in Problem 7.38 to write each in polar coordinates.

  1. \(2x-3y=6\)

  2. \(x^2+y^2=9\)

  3. \(25x^2+4y^2=100\)

  4. \(x = 4y^2\)

  5. \(xy=36\)

Problem 7.42.

Each of the following equations is written in the polar coordinate system. Use the relationships you found in Problem 7.39 to write each in Cartesian coordinates.

  1. \(r = 9 \cos (\theta)\)

  2. \(r^2 = 2 \sin (\theta)\)

  3. \(\dsp r = \frac{1}{3 \cos (\theta) - 2 \sin (\theta)}\)

  4. \(r = 8\)

  5. \(\dsp \theta = \frac{\pi}{4}\)

Problem 7.43.

Graph, in a polar coordinate system, the curve defined by each of the following equations.

  1. \(r=1+2 \cos (\theta)\)

  2. \(r = 2 + 2 \sin (\theta)\)

  3. \(r = 3 \sin (2 \theta)\)

  4. \(r = 2 \cos (3 \theta)\)

  5. \(r^2 = 4 \sin (2 \theta)\) Feel free to use graphing software to help.

Problem 7.44.

Find the points of intersection of the graphs of the given polar equations. (When I set them equal without graphing them, I always miss at least one point of intersection.)

  1. \(r = 4 - 4 \cos (\theta)\) and \(r = 4 \cos (\theta)\)

  2. \(r = 2 \cos (2 \theta)\) and \(r = \sqrt{3}\)

  3. \(r = \sin (\theta)\) and \(r = \sqrt{3} \cos (\theta)\)

  4. \(r = 4 + 2 \cos (\theta)\) and \(r = 5\)

  5. \(r = 1-2 \cos (\theta)\) and \(r = 1\)

Example 7.45.

Derive the formula for the arc length of a function starting with the distance formula. Derive the formula for the arc length of a planar curve defined by a parametric equation.

Example 7.46.

Compute the arc length (using the equation they will derive next) for the spiral \(r=\theta\) from \(\theta=0\) to \(\theta = 2\pi\text{.}\)

Problem 7.47.

If we display the independent variable \(\theta\text{,}\) then Cartesian and polar coordinates are related by \(x(\theta) = r(\theta) \cos (\theta)\) and \(y(\theta) = r(\theta) \sin (\theta)\text{.}\) Use the formula for the arc length of a parametric curve

\begin{equation*} \dsp L = \int_a^b \sqrt{\big(x'(t)\big)^2 + \big(y'(t)\big)^2} \; dt \end{equation*}

to derive the formula for the arc length of a curve in polar coordinates

\begin{equation*} \dsp L=\int_{\alpha}^{\beta} \sqrt{\big(r(\theta)\big)^2+\big(r'(\theta)\big)^2} \; d\theta. \end{equation*}
Problem 7.48.

Use the arc length formula from the previous problem and the half-angle identity to find the length of the cardioid defined by \(r = 3(1 + \cos (\theta))\text{.}\)

Problem 7.49.

Recall that \(\dsp {{dy} \over {\; dx}} = {{{dy} \over {d \theta}} \over {{dx} \over {d \theta}}}.\) Use this to find the tangent line to the curve in a polar plane \(r = 2 + 2 \sin (\theta)\) at \(\theta = \pi/4.\)

Example 7.50.

Compute the area of one of the four “petals” of \(r=3\sin(2\theta).\) Generalize to demonstrate a method for finding the area trapped inside polar functions.

Let \(r=f(\theta)\) be a continuous and non-negative function in the interval \([\alpha, \beta]\) and \(A\) be the area of the region bounded by the graph of \(f\) and the radial lines \(\theta = \alpha\) and \(\theta = \beta\text{.}\) To find \(A\text{,}\) partition the interval \([\alpha, \beta]\) into \(n\) subintervals \([\theta_0,\theta_1]\text{,}\) \([\theta_1, \theta_2]\text{,}\) \(\cdots\text{,}\) \([\theta_{n-1}, \theta_n]\) of equal lengths where \(\alpha = \theta_0 \lt \theta_1 \lt \cdots \lt \theta_n = \beta\text{.}\) Since the area of a circular sector of radius \(r\) and angle \(\theta\) is \(\dsp {1 \over 2}r^2 \theta\text{,}\) the area of each of our sectors is \(\dsp{1 \over 2} (\theta_i - \theta_{i-1})^2\) so the total area is approximately the sum of these sectors. Thus \(A = \displaystyle{\lim_{n \to \infty} {1 \over 2} \sum_{i=1}^n \big[f(\theta_i)\big]^2 \big(\theta_{i}-\theta_{i-1}\big) ={1 \over 2} \int_{\alpha}^{\beta} [f(\theta)]^2 d \theta}\text{.}\)

Problem 7.51.

Find the area of each region in a polar plane.

  1. the region bounded by the graph of \(r = -2 \cos (\theta)\)

  2. the region bounded by one loop of the graph of the equation \(r = 4 \cos (2 \theta)\)

  3. the region bounded by the inner loop of the curve \(r=2+3 \cos (\theta)\)

  4. the region bounded by the intersection of the two graphs \(r=4 \sin (2\theta)\) and \(r = 4 \cos (2 \theta)\)

  5. the region that lies inside the cardioid \(r = 2 - 2 \cos (\theta)\) and outside the circle \(r=1\)

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